Affine Primitive Groups and Semisymmetric Graphs

Aﬃne primitive groups and Semisymmetric graphs

Hua Han Zai Ping Lu∗ Center for Combinatorics Center for Combinatorics LPMC, Nankai University LPMC, Nankai University Tianjin 300071, P. R. China Tianjin 300071, P. R. China [email protected] [email protected]

Submitted: Jul 14, 2012; Accepted: May 21, 2013; Published: May 31, 2013 Mathematics Subject Classiﬁcations: 05C25, 20B25

Abstract In this paper, we investigate semisymmetric graphs which arise from aﬃne primitive permutation groups. We give a characterization of such graphs, and then construct an inﬁnite family of semisymmetric graphs which contains the Gray graph as the third smallest member. Then, as a consequence, we obtain a factorization of

the complete bipartite graph Kpspt ,pspt into connected semisymmetric graphs, where p is an prime, 1 6 t 6 s with s > 2 while p = 2.

Keywords: semisymmetric graph; normal quotient; primitive permutation group

1 Introduction

All graphs considered in this paper are assumed to be ﬁnite and simple with non-empty edge sets. For a graph Γ , denote by V Γ , EΓ and AutΓ the vertex set, edge set and automorphism group, respectively. A graph Γ is said to be vertex-transitive or edge-transitive if AutΓ acts transitively on V Γ or EΓ , respectively. A regular graph is called semisymmetric if it is edge-transitive but not vertex-transitive. For a graph Γ , an arc of Γ is an ordered pair (α, β) of two adjacency vertices. A graph Γ is called symmetric if it has no isolated vertices and AutΓ acts transitively on the set of arcs of Γ . The class of semisymmetric graphs was ﬁrst studied by Folkman [9], who possed several open problem. Afterwards, many authors have done much work on this topic, see [1, 2, 3, 7, 8, 10, 11, 12, 15, 16, 17, 18, 19] for references. In particular, lots of interesting examples of such graphs were found. For example, the Folkman graph on 20 vertices, the smallest semisymmetric graph, was constructed by Folkman [9]; the Gray graph, a

∗Supported in part by the NSF of China.

the electronic journal of combinatorics 20(2) (2013), #P39 1 cubic graph of order 54, was first observed to be semisymmetric by Bouwer [1]. In 1985, Iofinova and Ivanov [11] classified all bi-primitive cubic semisymmetric graphs and they proved that there are only five such graphs. In this paper, we consider the semisymmetric graphs whose automorphism group contains a subgroup inducing an affine primitive permutation group. For more information about groups of this kind, see [5, 6]. To state our result, we need to introduce several concepts and some notation. Let p be a prime and Fp the field of order p. Then, for an integer l > 1 and an irreducible subgroup H of the general linear group GL(l, p), all affine transformations τh,v l form a primitive subgroup X of AGL(l, p) (acting on the vectors of Fp), where h ∈ H, l l l h v ∈ Fp and τh,v is defined as Fp → Fp, u 7→ u + v. The above group is a split extension l of a regular normal subgroup {τ1,v | v ∈ Fp} by the subgroup H. For convenience, for a l subspace V of Fp, we set T(V ) = {τ1,v | v ∈ V } and denote by NH (V ) the subgroup of H l fixing V set-wise. Then X = T(Fp):H, and NH (V ) = NH (T(V )), the normalizer of T(V ) in H. For a graph Γ and a subgroup G 6 AutΓ , Γ is said to be G-vertex-transitive or G-edge-transitive if G is transitive on V Γ or EΓ , respectively. The graph Γ is called G-semisymmetric graphs if it is regular, G-edge-transitive but not G-vertex-transitive. Let Γ be a connected G-edge-transitive graph, where G 6 AutΓ . Assume that G is not transitive on V Γ . Then Γ is a bipartite graph with two parts, say U and W , which are the G-orbits on V Γ . Denote respectively by GU and GW the permutation groups induced by G on U and on W . Our main result deals with the case where one of GU and GW is an affine primitive permutation group.

l Theorem 1.1. Let X = T(Fp):H, where p is a prime, l > 1 and H is an irreducible subgroup of the general linear group GL(l, p). Then the following two statements are equivalent.

l (1) Fp has an s-dimensional subspace V for some integer 1 6 s < l such that |H : t NH (V )| = p for some integer 1 6 t 6 s. (2) There exists a semisymmetric graph Γ with bipartition V Γ = U ∪ W with one of GU and GW is permutation isomorphic to X for some edge-transitive subgroup G of AutΓ. Remark. Theorem 1.1 suggests the following interesting problem. Problem 1. Characterize irreducible subgroups of the general linear group GL(l, p) satisfying Theorem 1.1 (1). It is well-known that there are no semisymmetric graph of orders 16, 2p and 2p2, see [9]. Thus, for Theorem 1.1 (1), we have l > 3 and (p, l) 6= (2, 3).

2 Proof of Theorem 1.1

Assume that Γ is a G-edge-transitive but not G-vertex-transitive graph, where G 6 AutΓ . Then Γ is a bipartite graph with two parts, say U and W , which are the G-orbits on V Γ . the electronic journal of combinatorics 20(2) (2013), #P39 2 It follows that Γ is semiregular, that is, the vertices in a same bipartition subset have the same valency. For a given vertex α ∈ V Γ , the stabilizer acts transitively on Γ (α). Thus, if Γ is vertex-transitive then it must be symmetric. Take β ∈ Γ (α). Then each vertex of Γ can be written as αg or βg for some g ∈ G. Then, for two arbitrary vertices αg and βh, they hg−1 −1 are adjacent in Γ if and only if α and β are adjacent, i.e., hg ∈ GβGα. Moreover, it is well-known and easily shown that Γ is connected if and only if hGα,Gβi = G. Let Γ be a G-semisymmetric graph with two bipartition subsets U and W . Suppose that G has a subgroup R which is regular on both U and W . Take an edge {α, β} ∈ EΓ . Then each vertex in U (W , resp.) can be written uniquely as αx (βx, resp.) for some x ∈ R. Set S = {s ∈ R | βs ∈ Γ (α)}. Then αx and βy are adjacent if and only if yx−1 ∈ S. If R is abelian, then it is easily shown that αx 7→ βx−1 , βx 7→ αx−1 , ∀ x ∈ R is an automorphism of Γ , which leads to the vertex-transitivity of Γ , refer to [8, 14].

Lemma 2.1. Let Γ be a G-semisymmetric graph. Assume that G has an abelian subgroup which is regular on both parts of Γ . Then Γ is symmetric.

Let Γ be a G-semisymmetric graph. Suppose that G has a normal subgroup N which acts intransitively on at least one of the bipartition subsets of Γ . Then we define the 0 quotient graph ΓN to have vertices the N-orbits on V Γ , and two N-orbits ∆ and ∆ are 0 adjacent in ΓN if and only if some α ∈ ∆ and some β ∈ ∆ are adjacent in Γ . It is easy to see that the quotient ΓN is a regular graph if and only if all N-orbits have the same length. Let Γ be a finite connected G-semisymmetric graph with G 6 AutΓ . Take an edge {α, β} ∈ EΓ and let U = αG and W = βG be the two G-orbits on V Γ . Assume that G is unfaithful on U, and let K be the kernel of G acting on U. Then K is faithful on W , and each K-orbits on W contains at least two vertices. It follows that there are two distinct vertices in W which have the same neighborhood in Γ . Thus, as observed in [8], if any two distinct vertices in U have different neighborhoods in the quotient ΓK then Γ is semisymmetric and can be reconstructed from ΓK as follows. Construction 2.2. Let Σ be a bipartite graph with two bipartition subset U and W¯ such that m|W¯ | = |U| for integer m > 1. Define a bipartite graph Σ1,m with vertex ¯ set U ∪ (W ×Zm) such that α and (β, i) are adjacent if and only if {α, β} ∈ EΓ . For convenience, we set Σ1,1 = Σ.

For a group X, the socle of X, denoted by soc(X), is generated by all minimal normal subgroups of X. A permutation group is called quasiprimitive if each of its minimal normal subgroups is transitive.

Lemma 2.3. Let Γ be a ﬁnite connected G-semisymmetric graph with G 6 AutΓ . Take an edge {α, β} ∈ EΓ and set U = αG and W = βG. Assume that GU is quasiprimitive. Then either G is faithful on both U and W , or one of the following statements hold.

(1) Γ is isomorphic to the complete bipartite graph K|U|,|U|;

the electronic journal of combinatorics 20(2) (2013), #P39 3 (2) G is faithful on W but not faithful on U, GU ∼= GW¯ , and Γ is semisymmetric if further GU is primitive, where K is the kernel of G on U and W¯ is the set of K-orbits on W . Proof. To prove this lemma, we assume next that G is unfaithful on at least one of U and ∼ W and that Γ 6= K|U|,|U|. Since G 6 AutΓ , the kernel of G on one bipartition subset must be faithful on the other one. Then the above assumption implies that G is faithful on W . Thus G is unfaithful on U. Let K and W¯ be as in the lemma. Then GU ∼= G/K and G induces ¯ ∼ ¯ ¯ a subgroup G = G/K of AutΓK . Suppose that G is unfaithful on W . Then it follows ∼ ∼ K that ΓK = K|U|,|W¯ |, and so Γ = K|U|,|U| by noting that β ⊆ Γ (α) if β ∈ Γ (α), a contradiction. Thus G¯ is faithful on W¯ , so K is the kernel of G acting on W¯ , and hence GU ∼= G/K ∼= GW¯ . Note that each K-orbit on W has size at least 2, and that two vertices in a same K-orbit have the same neighborhood in Γ . Assume that GU is primitive. Then, since ∼ Γ 6= K|U|,|U|, any two distinct vertices in U have diﬀerent neighborhood. Thus there is no x ∈ AutΓ with U x = W , so Γ must be semisymmetric.

Corollary 2.4. Let G a finite group which acts faithfully and transitively on both nonempty ¯ ¯ sets U and W with |U| = m|W | for an integer m > 1. For α ∈ U and a Gα-orbit Θ on W¯ , define a bipartite graph Σ on U ∪ W¯ such that αg ∈ U and β¯ ∈ W¯ are adjacent if and only if β¯ ∈ Θg. If GU is primitive, then Σ1,m is a semisymmetric graph unless Θ = W¯ . Proof. Assume that GU is primitive and Θ 6= W¯ . By Lemma 2.3, it suffices to show that AutΣ1,m has an edge-transitive subgroup which fixes U and induces a permutation group U 1,m on U permutation isomorphic to G . Let Y = G×Zm. Define an action of Y on V Σ as follows:

g (x,i) gx ¯ (x,i) ¯x ¯ ¯ (α ) = α , (β, j) = (β , i + j), ∀g, x ∈ G, i, j ∈ Zm, β ∈ W. Then, under the above action, Y is a subgroup of AutΣ1,m as desired.

Remark. A graph is called edge-primitive if its automorphism group is primitive on its edge set. Using Corollary 2.4, we can construct examples of semisymmetric graphs from an edge-primitive graph of even valency, such as the complete graph K2l+1, the Perkel graph and etc., by taking U, W¯ and Θ respectively the edge set, vertex set and an orbit on W¯ of some edge-stabilizer of the edge-primitive graph. Here we pose the next interesting problem. Problem 2. Characterize or classify the primitive subgroups of Sn which have transitive permutation representations of degree properly dividing n.

Lemma 2.5. Let Γ be a connected G-semisymmetric graph with G 6 AutΓ . Take an edge {α, β} ∈ EΓ and set U = αG and W = βG. Assume that G is faithful on both U and W . Assume that GU is an aﬃne primitive permutation group and Γ is not a complete bipartite graph. Then either G is primitive on W , or Γ is semisymmetric.

the electronic journal of combinatorics 20(2) (2013), #P39 4 ∼ l Proof. Set N = soc(G). Then N = Zp for a prime p and integer l > 1. It is easily shown that G is primitive on W if and only if N is transitive on W . To ﬁnish the proof, in the following, we prove that Γ is semisymmetric if N is intransitive on W . Suppose that N is intransitive on W and Γ is symmetric. Then Nγ 6= 1 for any γ ∈ W . Let X be the set-wise stabilizer of U in AutΓ . Then G 6 X and |AutΓ : X| = 2. Note that XU is a primitive permutation group. By Lemma 2.3, since Γ is not a complete bipartite graph, we may assume that X is faithful on both U and W . Let M = soc(X). Take x ∈ AutΓ with αx = β and βx = α. Then U x = W and W x = U. Note that X is a primitive permutation group (on U) of degree pl. Then M is the unique minimal normal subgroup of X by the O’Nan-Scott Theorem, refer to [4, Theorem 4.1A]. Thus M x = M, it follows that M is transitive on both U and W . If X is of aﬃne type then M = N by [20, Proposition 5.1], so N is transitive on U, a contradiction. Then, by [13] and [20, Proposition 5.1], N < M and M is listed as follows: ∼ l (i) M = T , where (p, T ) is one of (11, PSL(2, 11)), (11, M11), (23, M23) and qd−1 ( q−1 , PSL(d, q)); or ∼ s (ii) M = T1× · · · ×Tt and N = (N ∩T1)× · · · ×(N ∩Tt), where Ti = Aps and N ∩Ti = Zp s for 1 6 i 6 t, p > 5 and st = l.

For (i) and (ii) with s = 1, the stabilizer Mα has order coprime to p, and so does for x Mα = Mβ, hence N ∩Mβ = 1, which contradicts that Nβ 6= 1. Thus (ii) occurs and s > 2, s ∼ so p > 5. It is easily shown that (Ti)α = Aps−1 and Mα = (T1)α× · · · ×(Tt)α. Then Mβ = x x x Mα = ((T1)α× · · · ×(Tt)α) = ((T1∩Mα)× · · · ×(Tt∩Mα)) = (T1)β× · · · ×(Tt)β. It implies x that {(T1)α, ··· , (Tt)α} = {(T1)β, ··· , (Tt)β} as all (Ti)α are simple and nonabelian. In ∼ particular, (Ti)β = Aps−1, so (Ti)α and (Ti)β are conjugate in Ti for 1 6 i 6 t. Thus Mα and Mβ are conjugate in M. Since M is transitive on W , there is γ ∈ W with Mα = Mγ. Then Nγ = N ∩ Mγ = N ∩ Mα = 1 as N is regular on U, again a contradiction. This completes the proof. Remark It is well-known that there are no symmetric graphs of order 2p and 2p2. Thus, ∼ l for Lemma 2.5, if N is intransitive on W then N = Zp for l > 3, which also follows from checking the irreducible subgroups of GL(2, p). Proof of Theorem 1.1. Let Γ be a semisymmetric graph with bipartition V Γ = U ∪ W satisfying Theorem 1.1 (2). Then Γ must be connected. Without loss of generality, we assume that GU is permutation isomorphic to X. By Lemma 2.3, G is faithful on W . Let K be the kernel of G acting on U and W¯ be the set of K-orbits on W , while K = 1 and ¯ W = W if G is faithful on U. Then, by Lemmas 2.3 and 2.5, ΓK is edge-transitive but ¯ ¯ ∼ not vertex-transitive. Let G be the subgroup of AutΓK induced by G. Then G = G/K, and G¯U is permutation isomorphic to X. Set N = soc(G¯). Suppose that N is transitive on W¯ . Then, since N is faithful ¯ ¯ on W , we have |W | = |N| = |U|. Thus K = 1 and Γ = ΓK , Γ is symmetric by ¯ ¯ Lemma 2.1, a contradiction. Then N is intransitive on W . Take an edge {α, β} of ΓK ¯ ¯ ¯ with α ∈ U and β ∈ W . Then each N-orbit on W has size |N:Nβ¯|= 6 1, and N has |W¯ | ¯ s |W¯ | r exactly orbits on W . Set |Nβ¯| = p and = p . Then 1 r s < l. Since |N:Nβ¯| |N:Nβ¯| 6 6 the electronic journal of combinatorics 20(2) (2013), #P39 5 ¯ ¯ ¯ G is transitive on W , we know that Gα is transitive on the set of N-orbits. Let B be ¯ ¯ ¯ r ¯ ¯ an N-orbit containing β. Then |Gα :(Gα)B| = p . Since Gα is maximal in G, we have ¯ ¯ ¯ ¯ ¯ ¯ ¯ Gα 6 NG(Nβ). So |Gα :NGα (Nβ)| > 1. Consider the set-wise stabilizer (Gα)B. For ¯ ¯gx ¯g g ∈ (Gα)B, noting that Nβ¯ is the kernel of N acting on B, we have β = β for x ∈ Nβ¯, −1 t ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ so gxg ∈ N ∩ Gβ = Nβ. Thus (Gα)B 6 NGα (Nβ). Set |Gα :NGα (Nβ)| = p . Then t ¯ ¯ r ¯U t > 1, and p divides |Gα :(Gα)B| = p , so t 6 r 6 s. Noting that G is permutation isomorphic to X, (1) of Theorem 1.1 follows. Now we assume that Theorem 1.1 (1) holds. To show (2), it suﬃces to construct a ¯ suitable semisymmetric graph. Set R = T(V )NH (V ). Let W be the set of right cosets l ¯ l−s+t of R in X, and let U be the set of vectors in Fp. Then |W | = p . Extend X to a permutation group on U ∪ W¯ such that XU = X and X acts on W¯ by the right multiplication on the right cosets of R in X. It is easily shown that X is faithful on both ¯ |V ||H| t ¯ U and W . Let Θ = {Rh | h ∈ H}. Then |Θ| = |R| = |H :NH (V )| = p < |W |. Deﬁne a bipartite graph Σ on U ∪ W¯ such that u ∈ U and β¯ ∈ W¯ are adjacent if and only if β¯ ∈ Θτ1,u . Then Σ is X-edge-transitive. Let m = ps−t. If m = 1 then, by Lemma 2.5, Σ is a semisymmetric graph as desired. If m > 1 then, by Corollary 2.4, Σ1,m is a semisymmetric graph as desired. This completes the proof.

3 Some examples

l Let X = T(Fp):H be a primitive permutation group satisfying Theorem 1.1 (1). Then, up to isomorphism, each graph satisfying Theorem 1.1 (2) can be constructed from an X-edge-transitive graph which is not a complete bipartite graph and has one bipartition l subset coinciding with the underlaying set of Fp. Let Σ be such an X-edge-transitive graph l with two bipartition subsets U = Fp and W . Then, for each β ∈ W , the stabilizer Xβ l l normalizes (T(Fp))β = T(V ), where V is an s-dimensional subspace of U and Xβ ∩T(Fp) = l l r T(V ). Thus Xβ 6 NX (T(V )) = T(Fp)NH (V ). Assume that T(Fp) has p orbits on W , t s−r and set |H :NH (V )| = p . Then 1 6 t 6 r 6 s < l, and |Xβ| = p |H|. We next consider one extreme case. Assume that Xβ = T(V ):NH (V ). Then r = t, and W may be identiﬁed with ∪h∈H {u+ h l V | u ∈ Fp} with the action of X on W as follows:

0 0 0 h τ1,u0 h 0 h hh 0 l 0 (u + V ) = (u + u ) + V ∀ u, u ∈ Fp, h, h ∈ H.

l h h l For each u ∈ U = Fp, set Θ(u) = {u + V | h ∈ H}. Then {Θ(u) | u ∈ Fp} is the set of H-orbits on W . Moreover, |Θ(0)| = pt < |W | = pl−s+t, and so |Θ(u)| < |W | for each u ∈ U. Thus Σ is isomorphic to one of the graphs constructed as follows. l Construction 3.1. Let p, l, H and V be as in Theorem 1.1 (1). Let U = Fp and h l W = ∪h∈H {u + V | u ∈ Fp}. For each u0 ∈ U, deﬁne a bipartite graph Γ (p, l, s, t; H, u0) on U ∪ W such that u ∈ U and u0 + V h0 ∈ W are adjacent if and only if

0 h0 h h u + V = u + u0 + V for some h ∈ H,

the electronic journal of combinatorics 20(2) (2013), #P39 6 i.e., 0 h−1 0 −1 u0 + (u − u ) ∈ V and h h ∈ NH (V ) for some h ∈ H. The next result follows from Lemmas 2.3 and 2.5, Corollary 2.4 and the above argu- ment.

Corollary 3.2. If there is a graph Σ = Γ (p, l, s, t; H, u0) as in Construction 3.1, then Σ1,ps−t is semisymmetric. Now we construct an inﬁnite family of semisymmetric graphs. Example 3.3. Let p be a prime, and let s and t be two integers with 1 6 t 6 s such that t l s > 2 if further p = 2. Let l = sp . Write Fp in a direct sum

pt l M Fp = Ui i=1

of s-dimensional subspaces. Without loss of generality, for each i, we take {eij | 1 6 j 6 s} as a basis of U, where eij is the column vector with the ((i − 1)s + j)-th entry equal to 1 and the other entries equal to zero. Let H be the subgroup of GL(l, p) ﬁxes the above decomposition. Then H ∼= GL(s, p)o Spt . Let V = U1 and Σ = Γ (p, l, s, t; H, u). Set G(p, s, t; u) = Σ1,m, where m = ps−t with Σ1,m = 1 while m = 1. Then we get a family

spt G = {G(p, s, t; u) | 1 6 t 6 s, p is a prime, (p, s) 6= (2, 1), u ∈ Fp } of semisymmetric graphs, and the following statements hold: (i) G(p, s, t; 0) has valency ps = ptps−t;

0 0 t (ii) G(p, s, t; u) = G(p, s, t; u ) if u − u ∈ Ui for some 1 6 i 6 p ; 0 (iii) G(p, s, t; ei1) = G(p, s, t; ei01) for i, i > 2; s s t (iv) G(p, s, t; e21) has valency p (p − 1)(p − 1);

Pk Pk t (v) G(p, s, t; a=1 uia ) = G(p, s, t; a=1 eia1) for 2 6 i1 < i2 < ··· ik 6 p and ui ∈ Ui \{0};

Pk s s k−1 pt−1 t (vi) G(p, s, t; i=2 ei1) has valency p (p − 1) (k−1 ), where 2 6 k 6 p . t Therefore the complete graph Kpspt ,pspt can be factorized into p connected semisymmetric graphs. In particular, K27,27 is the edge-disjoint union of three semisymmetric graphs of valency 3, 12 and 12, say, G(3, 1, 1; 0), G(3, 1, 1; e21) and G(3, 1, 1; e21 + e31). By [3], there is a unique cubic semisymmetric graph of order 54. Thus G(3, 1, 1; 0) is in fact the Gray graph. The smallest members of G have order 32, which are G(2, 2, 1; 0) and G(2, 2, 1; e21) and have valency 4 and 12, respectively. the electronic journal of combinatorics 20(2) (2013), #P39 7 It is easily shown that, for s > 2, we can get the same graphs as in Example 3.3 if replace H by H ∩ SL(l, p). This is also true for s = 1 unless p = 3. p p Lp Example 3.4. Let p be an odd prime. Write Fp in a direct sum Fp = i=1 Ui of 1- dimensional subspaces. Assume that ei ∈ Ui for each i, where ei is the column vector with the i-th entry equal to 1 and the other entries equal to zero. Let H be the subgroup ∼ p−1 of SL(p, p) ﬁxes the above decomposition. Then H = Zp−1:Ap. Let V = U1 and S(p, p; u) = Γ (p, p, 1, 1; H, u).

Then each S(p, p; u) is semisymmetric graph, and the following statements hold:

(i) S(p, p; 0) = G(p, 1, 1; 0) has valency p;

2 (ii) S(p, p; ei) = G(p, 1, 1; ei) has valency p(p − 1) , for p > 5 and 2 6 i 6 p;

(iii) S(3, 3; e2) and S(3, 3; e3) have valency 6, and G(3, 1, 1, e2) is the edge-disjoint union of these two graphs;

Pk Pk k−1 p−1 (iv) S(p, p; i=2 ei) = G(p, 1, 1; i=2 ei) has valency p(p − 1) (k−1) for k > 3.

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the electronic journal of combinatorics 20(2) (2013), #P39 9